Extreme Extension

4000ms

262144K

Description:

For an array $$$b$$$ of $$$n$$$ integers, the extreme value of this array is the minimum number of times (possibly, zero) the following operation has to be performed to make $$$b$$$ non-decreasing:

Select an index $$$i$$$ such that $$$1 \le i \le |b|$$$, where $$$|b|$$$ is the current length of $$$b$$$.
Replace $$$b_i$$$ with two elements $$$x$$$ and $$$y$$$ such that $$$x$$$ and $$$y$$$ both are positive integers and $$$x + y = b_i$$$.
This way, the array $$$b$$$ changes and the next operation is performed on this modified array.

For example, if $$$b = [2, 4, 3]$$$ and index $$$2$$$ gets selected, then the possible arrays after this operation are $$$[2, \underline{1}, \underline{3}, 3]$$$, $$$[2, \underline{2}, \underline{2}, 3]$$$, or $$$[2, \underline{3}, \underline{1}, 3]$$$. And consequently, for this array, this single operation is enough to make it non-decreasing: $$$[2, 4, 3] \rightarrow [2, \underline{2}, \underline{2}, 3]$$$.

It's easy to see that every array of positive integers can be made non-decreasing this way.

YouKn0wWho has an array $$$a$$$ of $$$n$$$ integers. Help him find the sum of extreme values of all nonempty subarrays of $$$a$$$ modulo $$$998\,244\,353$$$. If a subarray appears in $$$a$$$ multiple times, its extreme value should be counted the number of times it appears.

An array $$$d$$$ is a subarray of an array $$$c$$$ if $$$d$$$ can be obtained from $$$c$$$ by deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end.

Input:

The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10\,000$$$) — the number of test cases.

The first line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 10^5$$$).

The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le 10^5$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases doesn't exceed $$$10^5$$$.

Output:

For each test case, print a single integer — the sum of extreme values of all subarrays of $$$a$$$ modulo $$$998\,244\,353$$$.

Sample Input:

4
3
5 4 3
4
3 2 1 4
1
69
8
7264 40515 28226 92776 35285 21709 75124 48163

Sample Output:

Note:

Let $$$f(l, r)$$$ denote the extreme value of $$$[a_l, a_{l+1}, \ldots, a_r]$$$.

In the first test case,

$$$f(1, 3) = 3$$$, because YouKn0wWho can perform the following operations on the subarray $$$[5, 4, 3]$$$ (the newly inserted elements are underlined):
$$$[5, 4, 3] \rightarrow [\underline{3}, \underline{2}, 4, 3] \rightarrow [3, 2, \underline{2}, \underline{2}, 3] \rightarrow [\underline{1}, \underline{2}, 2, 2, 2, 3]$$$;
$$$f(1, 2) = 1$$$, because $$$[5, 4] \rightarrow [\underline{2}, \underline{3}, 4]$$$;
$$$f(2, 3) = 1$$$, because $$$[4, 3] \rightarrow [\underline{1}, \underline{3}, 3]$$$;
$$$f(1, 1) = f(2, 2) = f(3, 3) = 0$$$, because they are already non-decreasing.

So the total sum of extreme values of all subarrays of $$$a = 3 + 1 + 1 + 0 + 0 + 0 = 5$$$.

Informação

Provedor Codeforces

Código CF1603C